Drawing Diagrams

Date: 08/02/98 at 06:51:18
From: Emma
Subject: Drawing diagrams
Hi,
In my extension maths class we have been given some problems to work
on. I think that I will be able to solve them; however, at the moment I
cannot draw them out. I have spent hours trying to draw them out on
paper and with my computer program Cabri. I would be grateful if you
could help draw them out for me. Is there a way to put
it on a web page where I can go and see how the correct diagrams would
look?
The first one I am having trouble is about congruent triangles.
In a quadrilateral ABCD, BC = AD, angle ABC = 150 degrees and angle
BAD = 70 degrees. E is a point on the diagonal BD such that angle
EAD = 20 degrees. It is also given that angle BDA = 30 degrees. Prove
that CE = BD.
The second one is: In a quadrilateral ABCD, AD is parallel to BC and
angle ABD = 45 degrees. X and Y are the feet of the perpendiculars
dropped from D on the sides AB and BC respectively, so that X is
between A and B and Y is between B and C. The line YX meets and
extended line DA at point in Z. Prove that ZD = YD.
All I need really is the correct diagrams. Maybe with your more
expansive maths knowledge you can see how to draw these diagrams
correctly. Thanks for you time!

Date: 08/02/98 at 23:32:24
From: Doctor Jaffee
Subject: Re: Drawing diagrams
Hi Emma,
It is possible to put diagrams on a Web site, but we'll start with a
verbal explanation that may help.
First of all, in problem 1, I would suggest that you start with a 150
degree angle and label the vertex B. Pick a point on one ray and call
it A. Next draw an 80 degree angle, ABX. (Draw the angle in the
interior of the 150 degree angle and I'll explain the 80 degrees
later).
Next draw angle BAE measuring 50 degrees so that E is on the ray BX.
Follow that with the angle EAD measuring 20 degrees so that D is on
the ray BX.
You can now locate the point C on the original 150 degree angle by
measuring the length of AD and marking off that distance on BC.
Finally, connect C and D and you have your diagram.
Now, when I originally made the drawing myself, I guessed at the
measure of angle ABD, but after I was finished I was able to calculate
that it had to be 80 degrees. So, I started over and redrew the
diagram with that 80 degree angle.
My guess is that one of the reasons you are having so much trouble
with these diagrams is that you are trying to draw them too exactly.
Try just drawing approximations and labelling the sides and angles to
indicate measures. Then as you learn more about the diagram, you can
revise it and get a more accurate drawing. I know this technique has
helped a lot of students at the school where I teach.
Let's take a look at the second one. Start off by drawing two
horizontal lines, the lower one being the line AE. Locate B on the
upper line a little to the right of A, then draw the 45 degree angle
ABD such that D is on AE. Locate C on the upper line so that is to the
right of D. Now if you draw DY perpendicular to BC with Y on the line
BC, Y will be between B and C.
Furthermore, if you draw DX perpendicular to AB with X on AB, X will
be between A and B. Finally, the line YX will intersect the line AD at
Z, and Z will not be between A and D.
I hope these verbal explanations help. Now that you have read through
them, here are the accompanying diagrams to help you with anything
that might not have been clear.
Question 1:
Question 2:
- Doctor Jaffee, The Math Forum
Check out our web site! http://mathforum.org/dr.math/